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Since the IDs of Tumblr posts are globally unique, I would expect there to be a way to get the full URL (someblog.tumblr.com/post/POST_ID) when all I have is POST_ID. Is there indeed a way?
I would also like this to work for blogs using a custom domain.
For blogs on tumblr.com, you should probably be able to use advanced Google search parameters. To wit:
site:tumblr.com inurl:{POST_ID}
My tests were inconclusive. Much of the time Google returned nothing, but other times it returned exactly one result.
Hmmm...it looks like maybe inurl: is deprecated. I had much better results with:
site:tumblr.com/post/{POST_ID}
For blogs with custom domains, you're probably limited to using inurl:.